Free Fall and Projectile Motion

Objects in free fall accelerate at g ≈ 9.8 m/s² downward. Projectile motion combines horizontal uniform motion with vertical free fall.

Free Fall

In free fall (ignoring air resistance), all objects accelerate at $g \approx 9.81 \text{ m/s}^2$ downward.

Free Fall Equations

v = gt, \quad h = \dfrac{1}{2}gt^2

Starting from rest: v = velocity after time t, h = height fallen.

Projectile Motion

A projectile has:

Horizontal: constant velocity $v_x = v_0\cos\theta$
Vertical: free fall with initial velocity $v_y = v_0\sin\theta$

Range of a Projectile

R = \dfrac{v_0^2 \sin 2\theta}{g}

Maximum range occurs at θ = 45°.

Object dropped from rest, falls 45 m. Find time and speed of impact.

$t = \sqrt{2h/g} = \sqrt{90/9.8} \approx 3.03 \text{ s}$

$v = gt = 9.8 \times 3.03 \approx 29.7 \text{ m/s}$

Real world: Dropping a phone off a balcony

A phone is accidentally knocked off a 5 m high balcony. How long before it hits the ground (g = 9.81 m/s²)?

$h = \dfrac{1}{2}gt^2 \implies t = \sqrt{\dfrac{2 \times 5}{9.81}} \approx 1.01 \text{ s}$

It hits the ground at: $v = 9.81 \times 1.01 \approx 9.9 \text{ m/s}$ — about 36 km/h. That is why screen protectors matter!

Real world: Skydiver before opening the parachute

A skydiver jumps from 4000 m and free-falls for 30 seconds (ignoring air resistance). Speed reached:

$v = g \times t = 9.81 \times 30 \approx 294 \text{ m/s}$

In reality air resistance limits this to around 55 m/s (terminal velocity) — but the free-fall model gives the upper bound.

Real world: Football kicked straight up

A player kicks a ball straight up at 15 m/s. How high does it go (g = 9.81 m/s²)?

$v^2 = u^2 - 2gh \implies 0 = 225 - 2 \times 9.81 \times h \implies h \approx 11.5 \text{ m}$

Tip

Where do you see this in real life? Engineers use free-fall equations to design safety airbags, helmet standards, and falling-object barriers on construction sites. Astronauts in orbit are actually in continuous free fall — they just keep missing the Earth!

Practice Problems

A stone falls from rest for 4 s. How far does it fall (g = 10 m/s²)?

Show answer

h = ½ × 10 × 16 = 80 m